Dtjytyk

Why do passenger jet manufacturers design their planes with stall prevention systems?

Why do newer 737s use two different styles of split winglets?

Simplify an interface for flexibly applying rules to periods of time

Do the common programs (for example: "ls", "cat") in Linux and BSD come from the same source code?

What does 高層ビルに何車線もの道路。mean?

What is the purpose or proof behind chain rule?

Is a party consisting of only a bard, a cleric, and a warlock functional long-term?

Non-trivial topology where only open sets are closed

Most cost effective thermostat setting: consistent temperature vs. lowest temperature possible

Why one should not leave fingerprints on bulbs and plugs?

Is "upgrade" the right word to use in this context?

Fastest way to pop N items from a large dict

Instead of a Universal Basic Income program, why not implement a "Universal Basic Needs" program?

Meme-controlled people

How are passwords stolen from companies if they only store hashes?

"of which" is correct here?

This word with a lot of past tenses

Happy pi day, everyone!

Bach's Toccata and Fugue in D minor breaks the "no parallel octaves" rule?

When to use a slotted vs. solid turner?

What is "focus distance lower/upper" and how is it different from depth of field?

Aluminum electrolytic or ceramic capacitors for linear regulator input and output?

How could a scammer know the apps on my phone / iTunes account?

Brexit - No Deal Rejection

Rudin Functional analysis, theorem 4.12 corollary (b)

Theorem 1.24 Rudin, Functional analysis (first half)Theorem 1.14 (b) Rudin functional analysis, few clarifications.Rudin functional analysis theorem 2.5.Rudin functional analysis, theorem 2.11 (Open mapping theorem)Rudin functional analysis, theorem 3.23. Finite intersection propertyRudin Functional Analysis Theorem 3.25Rudin functional analysis, theorem 3.27Rudin functional analysis theorem 3.28, application of Reisz representation theorem.Rudin functional analysis, theorem 4.9 part (b)Rudin functional analysis theorem 4.13, (a) and (b)

0

$begingroup$

Suppose $X$ and $Y$ are Banach spaces, and $T in mathcal{B}(X,Y)$ Then
$$
mathcal{N}(T^*) = mathcal{R}(T)^{perp} ;;text{and};; mathcal{N}(T) = ^{perp}mathcal{R}(T^*)
$$

The corollary in the title

b) $mathcal{R}(T)$ is dense in $Y$ iff $T^*$ is one-to-one

The proof is really short but there's a specific bit I don't get (fully at least)

$mathcal{R}(T)$ is dense in $Y$ iff $mathcal{R}(T)^{perp} = left{ 0right}$

I've tried to prove this bit by exercise, and I think the most promosing was to show that

$$
^{perp}(mathcal{R}(T)^{perp}) = Y iff mathcal{R}(T) = left{0right}
$$

The reason of the equality on the left of the double implication is because of theorem 4.7, which states that the orthogonal of the orthogonal of a set is the norm closure of a set.

However I'm not entirely sure how to exploit this, can you help?

functional-analysis proof-explanation normed-spaces topological-vector-spaces duality-theorems

share|cite|improve this question

edited Mar 11 at 11:51

user8469759

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

$endgroup$

add a comment | 

0

$begingroup$

Suppose $X$ and $Y$ are Banach spaces, and $T in mathcal{B}(X,Y)$ Then
$$
mathcal{N}(T^*) = mathcal{R}(T)^{perp} ;;text{and};; mathcal{N}(T) = ^{perp}mathcal{R}(T^*)
$$

The corollary in the title

b) $mathcal{R}(T)$ is dense in $Y$ iff $T^*$ is one-to-one

The proof is really short but there's a specific bit I don't get (fully at least)

$mathcal{R}(T)$ is dense in $Y$ iff $mathcal{R}(T)^{perp} = left{ 0right}$

I've tried to prove this bit by exercise, and I think the most promosing was to show that

$$
^{perp}(mathcal{R}(T)^{perp}) = Y iff mathcal{R}(T) = left{0right}
$$

The reason of the equality on the left of the double implication is because of theorem 4.7, which states that the orthogonal of the orthogonal of a set is the norm closure of a set.

However I'm not entirely sure how to exploit this, can you help?

functional-analysis proof-explanation normed-spaces topological-vector-spaces duality-theorems

share|cite|improve this question

edited Mar 11 at 11:51

user8469759

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

$endgroup$

add a comment | 

$begingroup$

Suppose $X$ and $Y$ are Banach spaces, and $T in mathcal{B}(X,Y)$ Then
$$
mathcal{N}(T^*) = mathcal{R}(T)^{perp} ;;text{and};; mathcal{N}(T) = ^{perp}mathcal{R}(T^*)
$$

The corollary in the title

b) $mathcal{R}(T)$ is dense in $Y$ iff $T^*$ is one-to-one

The proof is really short but there's a specific bit I don't get (fully at least)

$mathcal{R}(T)$ is dense in $Y$ iff $mathcal{R}(T)^{perp} = left{ 0right}$

I've tried to prove this bit by exercise, and I think the most promosing was to show that

$$
^{perp}(mathcal{R}(T)^{perp}) = Y iff mathcal{R}(T) = left{0right}
$$

The reason of the equality on the left of the double implication is because of theorem 4.7, which states that the orthogonal of the orthogonal of a set is the norm closure of a set.

However I'm not entirely sure how to exploit this, can you help?

functional-analysis proof-explanation normed-spaces topological-vector-spaces duality-theorems

share|cite|improve this question

edited Mar 11 at 11:51

user8469759

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

$endgroup$

share|cite|improve this question

edited Mar 11 at 11:51

user8469759

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

share|cite|improve this question

edited Mar 11 at 11:51

user8469759

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

asked Mar 11 at 10:24

user8469759user8469759

1,5581618

1 Answer
1

active

oldest

votes

0

$begingroup$

Here is another approach:

Suppose $R(T)$ is dense. Any linear functional that vanishes on $R(T)$ must vanish on the entire space by continuity considerations. That is, $R(T)^perp$ consists only of the zero functional.

Conversely suppose $R(T)$ is not dense. Then $overline{R(T)}$ is a proper closed subspace of $Y$ and there exists (per e.g. Hahn-Banach) a nontrivial linear function $phi$ that vanishes on $overline{R(T)}$, meaning $R(T)^perp not= {0}$.

share|cite|improve this answer

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267

$endgroup$

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3143526%2frudin-functional-analysis-theorem-4-12-corollary-b%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

0

$begingroup$

Here is another approach:

Suppose $R(T)$ is dense. Any linear functional that vanishes on $R(T)$ must vanish on the entire space by continuity considerations. That is, $R(T)^perp$ consists only of the zero functional.

Conversely suppose $R(T)$ is not dense. Then $overline{R(T)}$ is a proper closed subspace of $Y$ and there exists (per e.g. Hahn-Banach) a nontrivial linear function $phi$ that vanishes on $overline{R(T)}$, meaning $R(T)^perp not= {0}$.

share|cite|improve this answer

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267

$endgroup$

add a comment | 

0

$begingroup$

Here is another approach:

Suppose $R(T)$ is dense. Any linear functional that vanishes on $R(T)$ must vanish on the entire space by continuity considerations. That is, $R(T)^perp$ consists only of the zero functional.

Conversely suppose $R(T)$ is not dense. Then $overline{R(T)}$ is a proper closed subspace of $Y$ and there exists (per e.g. Hahn-Banach) a nontrivial linear function $phi$ that vanishes on $overline{R(T)}$, meaning $R(T)^perp not= {0}$.

share|cite|improve this answer

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267

$endgroup$

add a comment | 

$begingroup$

Here is another approach:

Suppose $R(T)$ is dense. Any linear functional that vanishes on $R(T)$ must vanish on the entire space by continuity considerations. That is, $R(T)^perp$ consists only of the zero functional.

Conversely suppose $R(T)$ is not dense. Then $overline{R(T)}$ is a proper closed subspace of $Y$ and there exists (per e.g. Hahn-Banach) a nontrivial linear function $phi$ that vanishes on $overline{R(T)}$, meaning $R(T)^perp not= {0}$.

share|cite|improve this answer

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267

$endgroup$

share|cite|improve this answer

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267

answered Mar 11 at 11:31

Umberto P.Umberto P.

39.9k13267